System and method for printing semiconductor patterns using an optimized illumination and reticle

ABSTRACT

A system and method is described for lithographically printing patterns on a semiconductor using combinations of illumination and mask patterns which are optimized together to produce the desired pattern. The method of optimizing both illumination and mask pattern allows the development of mask patterns that are not constrained by the geometry of the desired pattern to be printed. Thus, the method provides high quality images even when the desired printed patterns have critical dimensions that approach the resolution limits of a lithographic system. The resulting mask patterns using the method do not obviously correspond to the desired patterns to be printed. Such masks may include phase-shifting technology that use destructive interference to define dark areas of the image and are not constrained to conform to the desired printed pattern.

FIELD OF THE INVENTION

The present invention relates generally to lithographic printing offeatures for forming integrated circuit (IC) patterns on a semiconductorchip, and more particularly to a method for selecting and usingcombinations of illumination source characteristics and diffractingshapes on a reticle mask in order to project and print an image on asemiconductor wafer that substantially matches the shape of the desiredIC patterns with minimal distortion.

BACKGROUND OF THE INVENTION

Many methods have been developed to compensate for the image degradationthat occurs when the resolution of optical lithography systemsapproaches the critical dimensions (CD's) of desired lithographicpatterns that are used to form devices and integrated circuits (IC's) ona semiconductor chip. Critical dimension (CD) refers to the feature sizeand spacing between features and feature repeats (pitch) that arerequired by the design specifications and are critical for the properfunctioning of the devices on a chip. When the CD's of a desired ICpattern approach the resolution of a lithographic system (defined as thesmallest dimensions that can be reliably printed by the system), imagedistortions becomes a significant problem. Today the limited resolutionof lithography tools poses a key technical challenge in IC manufacture,and this difficulty will increase in the future as critical dimensionsbecome increasingly smaller. In order to make the manufacture of futureIC products feasible, lithography tools will be required to achieveadequate image fidelity when the ratio of minimum CD to resolution ofthe lithographic system is very low.

The resolution ρ of a lithographic system can be described by theequation: $\begin{matrix}{{\rho = \frac{k\quad \lambda}{NA}},} & \lbrack 1\rbrack\end{matrix}$

where ρ is the minimum feature size that can be lithographicallyprinted, NA (numerical aperture) is a measure of the amount of lightthat can be collected by the lens, and λ is the wavelength of the sourcelight. This equation expresses the concept that the smallest featuresize that can be printed is proportional to the wavelength of the lightsource, and that the image fidelity is improved as diffracted light iscollected by the lens over a wider range of directions. Although alarger NA permits smaller features to be printed, in practice NA islimited by depth-of-focus requirements, by polarization and thin-filmeffects, and by difficulties in lens design. The so-called k-factorrepresents aspects of the lithographic process other than wavelength ornumerical aperture, such as resist properties or the use of enhancedmasks. Typical k-factor values in the prior art range from about 0.7 to0.4. Because of limitations in reducing wavelength λ or increasingnumerical aperture NA, the manufacture of future IC products having verysmall CD's will require reducing the k-factor, for example, to the range0.3-0.4 or smaller, in order to improve the resolution of thelithographic processes.

The basic components of a projection lithographic system are illustratedin FIG. 1. An illumination source 110 provides radiation thatilluminates a mask 120, also known as a reticle; the terms mask andreticle may be used interchangeably. The reticle 120 includes featuresthat act to diffract the illuminating radiation through a lens 140 whichprojects an image onto an image plane, for example, a semiconductorwafer 150. The aggregate amount of radiation transmitted from thereticle 120 to the lens 140 may be controlled by a pupil 130. Theillumination source 110 may be capable of controlling various sourceparameters such as direction and intensity. The wafer 150 typicallyincludes a photoactive material (known as a resist). When the resist isexposed to the projected image, the developed features closely conformto the desired pattern of features required for the desired IC circuitand devices.

The pattern of features on the reticle 120 acts as a diffractingstructure analogous to a diffraction grating which transmits radiationpatterns that may interfere constructively or destructively. Thispattern of constructive and destructive interference can be convenientlydescribed in terms of a Fourier transform in space based on spacing ofthe features of the diffraction grating (or reticle 120). The Fouriercomponents of diffracted energy associated with the spatial frequenciesof the diffracting structure are known in the art as diffracted orders.For example, the zeroth order is associated with the DC component, buthigher orders are related to the wavelength of the illuminatingradiation and inversely related to the spacing (known as pitch) betweenrepeating diffracting features. When the pitch of features is smaller,the angle of diffraction is larger, so that higher diffracted orderswill be diffracted at angles larger than the numerical aperture of thelens.

A diagram can be constructed in direction space to indicate thediffracted orders that can be collected by a lithographic system that isbased on repeating dimensions of a desired pattern. For example, thepattern illustrated in FIG. 4 can be represented by a unit cell as inFIG. 2. The pattern has a horizontal repeat dimension 203, and astaggered pitch indicated by the diagonal repeat dimension 205(alternatively indicated by the vertical pitch 201). Assuming that thisunit cell is repeated in a diffraction grating and illuminated by anon-axis beam, the diffracted orders can be illustrated in directionspace as indicated in FIG. 3. The position of a diffracted order (points300-326) is plotted as the projection of the beam diffracted at an angleθ from the on-axis beam. The distance of a non-zero order from thecenter of the direction space diagram 300 (which represents the positionof the zeroth order and is also the direction of the on-axis beam) isplotted as the sine of θ which is the ratio of the wavelength of theillumination divided by the repeat distance. For example, the +2 orderrepresented by the horizontal repeat distance 203 is represented by thepoint 301 and the −2 order is represented by the point 310. Similarly,points 305 and 319 represent the +2 and −2 orders based on the verticalrepeat distance 201. Other orders are diffracted both horizontally andvertically, such as order 308, denoted as the {−1, +1} order. Forreference, the numerical aperture (NA) 350 of the lens is also plotted.The only orders collected by the lens are 300, 301, 310, 303, 308, 313,and 312. Note that the amplitudes of a wave front diffracted by areticle will be dependent on both the illumination amplitude and thediffractive properties of the mask.

Off-axis illumination has been known in the art as a technique used toimprove resolution. Although off-axis illumination causes asymmetry inthe projected image, the asymmetry caused by the off-axis illuminationcan be corrected by illuminating from mirrored directions. Thistechnique is often used in the prior art, for example, by using anannular source configuration.

The intensity contours of light projected by the lens can departsignificantly in shape from those of the input mask pattern. Twodimensional (2D) patterns have multiple critical dimensions that must bemet, thus exacerbating the problem of achieving image fidelity.Moreover, with all but the simplest shapes, the errors in the differentcritical dimensions that comprise the printed pattern are unequal,making it impossible to correct the errors with an exposure adjustment.Quite often such unequal dimensional distortions fall into the broadcategory of “line-shortening”. For example, patterns such as in FIG. 4(for example, an isolation level of a dynamic random access memory(DRAM) design) or as in FIG. 14 (for example, the capacitor level of aDRAM design) are prone to line-shortening. In the pattern of FIG. 4, therectangular features, have width 401 equal to the basic dimensional unitof the cell F. The rectangular features represent regions wherephotoactive material (resist) should be retained after the printedpattern is developed. The vertical spacing 402 is also equal to F, andthe length 405 is equal to 6.5 F. The desired horizontal spacing 408between the tips of the rectangles is 1.5 F. However, when the k-factoris small, the contrast across the tips is small, and in order toadequately resolve the tips of the rectangles, it is necessary to printthe rectangles at a length shorter than the desired 6.5 F.

In addition, at small k-factor, the low contrast of the projected imagesmagnifies the dimensional errors that arise from random variations inthe patterning process. This can cause prohibitive sensitivities to suchimperfections as non-uniform substrate reflectivity, mask dimensionalinaccuracy, illumination nonuniformity, defocus, stray light, andresidual lens aberrations.

Many methods have been developed to reduce these problems. A summary ofthese prior art methods is briefly described below.

Many enhancement techniques adjust the shapes of mask features tocompensate the distortions that arise at small k-factor, as discussed inL. W. Liebmann et al., “Optical proximity correction: a first look atmanufacturability,” in SPIE Proceedings, Vol. 2322—14th Annual Symposiumon Photomask Technology and Management, (Society of Photo-OpticalInstrumentation Engineers, 1994), pages 229-238. The technique ofaltering the reticle mask shapes (for example, by widening the maskshapes at the tips of line features, or by lengthening the features) isreferred to as biasing. For example, FIG. 4A illustrates a mask biasedwith hammerhead shapes 420 to compensate for line-shortening of thepattern in FIG. 4. In some cases, however, this not only fails toaddress the problem of poor contrast, it actually exacerbates it, i.e.biasing mask features can actually degrade contrast to the point ofbeing counterproductive. In cases such as the pattern in FIG. 4, thecontrast across the tips is poor even when the rectangles print withconsiderable shortening; i.e. considerable light spills between adjacenttips in the blurred image, even though shortening draws the tips apart.When the mask rectangles are biased with hammerhead shapes 420 as inFIG. 4A in order to compensate for line-shortening, the contrast in thegaps of the image degrades further because poorly resolved light fromthe hammerheads 420 spills into gaps 409. Contrast in the gaps 409 issimilarly degraded if the mask rectangles are lengthened in an effort tocompensate for line shortening in the printed patterns, because blurringis worse when the separating gaps are biased to be narrower.

Computer algorithms are known which can provide appropriate adjustmentof mask shapes to compromise between such conflicting effects (see, forexample, O. W. Otto et al., “Automated optical proximity correction—arules-based approach,” in SPIE Proceedings, Vol. 2197—OpticalMicrolithography VII (Society of Photo-Optical InstrumentationEngineers, 1994), pages 278-293). However, these algorithms are onlyable to provide a very limited benefit when different aspects of imagequality require that the shapes be perturbed in opposite directions, aswith line shortening. In general, image enhancement techniques workpoorly when geometric constraints that are inherent to the desiredcircuit pattern yield contradictory requirements for optimizing theshape and/or position of these patterns on the mask. For example, theclose packing of patterns such as in FIG. 4 or FIG. 14 causes anintrinsic loss in contrast when mask features are biased to achieve thedesired critical dimensions (CD's) of the desired image.

Another class of enhancement techniques improves contrast in the imageby shifting the phase of the light projected from the mask. This doesnot directly address the above-mentioned intrinsic geometric conflictsin certain circuit patterns, but it does reduce their severity byreducing image blur. One source of image blur is caused by the limitedresolution of lithography lenses, which washes out the sharp transitionin transmittance between mask features, blurring it over a distancedefined by the lens resolution.

One enhancement technique (known as “phase-shifting chrome” or“attenuated phase shift”) improves image sharpness by augmenting therate of change in illumination amplitude across the edge of maskfeatures. This is achieved by using phase-shifting material of slightlynegative transmittance for dark areas of the pattern, rather than theconventional material of zero transmittance, for example, as describedin T. Terasawa et al., “Imaging characteristics of multi-phase-shiftingand halftone phase-shifting masks,” Japanese J. Appl. Phys. Part 1, Vol.30, no.11B (1991), pages 2991-2997. Phase shifting increases the slopeof illumination intensity at the edges of image features since thetransmitted electric field makes a transition (see, for example electricfield amplitude 160 in FIG. 1A) from unity to a value less than zero(see, for example electric field amplitude 160 in the dark region 199 inFIG. 1A); the slope in the image intensity across the edge of featuresin the image is increased accordingly. However, the steepness of theslope across the edges of image features is limited by the requirementthat the negative electric field amplitude 160 transmitted to areas ofthe image that are intended to be dark areas 199 not have sufficientintensity 170 to print the dark areas 199 (FIG. 1A) as if they werebright areas 190. (For purposes of discussion, it is hereafter assumedthat the photoresist is a positive resist, which is most commonly usedin the art. In the case of a negative resist, dark image areas would besubstituted for bright areas and vice versa.) Thus, while phase shiftingimproves contrast, the improvement can be inadequate in certain cases.As previously discussed, certain patterns are limited by intrinsicgeometric constraints in which correction of dimensional errors can onlybe made at the expense of degraded contrast. Phase-shifting reduces theimpact of these pattern conflicts, but does not eliminate them. The sameconclusion applies when negative electric field amplitude is provided bya thin rim of transparent phase shifting material as discussed in A.Nitayama et al., “New phase-shifting mask with self-aligned phaseshifters for quarter micron photolithography,” in 1989 InternationalElectron Devices Meeting—Technical Digest (Cat. 89CH2637-7) (Washington,D.C.: IEEE, 1989), pages 57-60.

The so-called alternating-phase-shift technique (for example, asdiscused in M. D. Levenson, N. S. Viswanathan, and R. A. Simpson,“Improving Resolution in Photolithography with a Phase-Shifting Mask,”IEEE Transactions on Electron Devices, Vol. ED-29, no. 12 (1982), pages1828-1836) achieves further contrast improvement by successivelyshifting the phase of adjacent bright features between 0° and 180°. Inthis way the contrast of illumination intensity across the edge of imagefeatures is further increased in comparison to either conventional masksor phase-shifting chrome. However, as with phase-shifting chrome, thealternating-phase-shift technique does not directly address theabove-mentioned intrinsic geometric constraints of common 2D patterns,though it can further reduce their severity. In addition, with some 2Dcircuit layouts it is impossible to tile every mask feature with a phasethat is opposite to the phase of all neighboring features, meaning thatlithographic performance will be gated by the unimproved transitionsthat separate features having the same phase.

Moreover, the alternating-phase-shift technique often adds unwantedfeatures to image patterns. This occurs when circuit shapes are laid outin such a way that the desired alternation in phase can only be achievedby introducing artificial 0° to 180° mask transitions which print asunwanted patterns. For example, when opposite phases are applied tobright regions that pass in close proximity to one another at a certainpoint on the mask, the phase must make such an unwanted transition ifthe bright regions are connected together elsewhere in the mask pattern.Such unwanted phase transitions will print as a dark fringe within thenominally bright connecting area, and must be trimmed away using asecond exposure. It has also been suggested that unwanted masktransitions might be blunted below the threshold of printability throughuse of intermediate-phased regions, grading the transition in stagesfrom 0° to 180° along the connecting regions. However, this gives riseto a phase tilt along the mask, which in turn causes very strongshifting of the image when focus fluctuates. For this reasonintermediate phases are not often employed.

It is known that the benefits of a continuously varying phase cansometimes be achieved by tilting the light beam which illuminates themask (for example, see N. Shiraishi et al., “New imaging technique for64M DRAM,” in SPIE Proceedings, Vol. 1674—Optical Microlithography V(Society of Photo-Optical Instrumentation Engineers, 1992), pages741-752; M. Noguchi et al., “Sub-half-micron lithography system withphase-shifting effect,” in SPIE Proceedings Vol.1674—OpticalMicrolithography V (Society of Photo-Optical Instrumentation Engineers,1992), pages 92-104). With many patterns the tilt can be adjusted insuch a way that the change in tilt phase along the mask causes theillumination at successive critical features to alternate betweenpositive and negative phase. Moreover, where successive features areconnected by orthogonal shapes, the phase makes a smooth transition from0° to 180° along these connecting shapes. The above-mentioned focussensitivity which such phase tilts can cause is avoided by illuminatingthe mask symmetrically from mirrored directions. In lowest order thefocus sensitivities from the different directions then cancel.

Methods are known for selecting the illumination directions incident ona given mask in ways that maximize the slope of image features, and thatminimize CD nonuniformity between different features throughsuperposition of multiple illumination directions (for example, see U.S.Pat. No. 5,680,588 entitled “Method and system for optimizingillumination in an optical photolithography projection imaging system”issued to A. E. Rosenbluth and J. Gortych on Oct. 21, 1997). This isreferred to as “source optimization”. However, as with the imageenhancement techniques described above, the benefit from optimizing theillumination in this way is limited. The optimized source achieves CDuniformity by balancing the differing bias effects of multipleillumination directions. Unfortunately, when bias effects are severe,for example when the geometric constraints of the pattern result in lineshortening, such balancing usually requires contributions from imagecomponents produced by particular illumination directions that have lowcontrast.

Accordingly, there is a need for a technique for enhancing image qualitythat is not so strongly limited by the intrinsic geometric constraintsof the pattern layout.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a method foroptimally choosing illumination distribution and reticle mask featuresso that the number of adjustable degrees of freedom per resolutionelement is significantly increased.

It is a further object of the present invention to significantly reducephenomena, such as line shortening, that are due to constraints inherentin the geometry of the desired wafer patterns.

It is a further object of the present invention to obtain optimalcombinations of illumination and mask patterns without requiring thatdiffracted wave fronts collected by the lens be constrained to besymmetrical.

It is a further object of the present invention to obtain optimal maskpatterns that are not constrained to conform to the basic layout of thedesired target wafer patterns.

The present invention addresses the above-described objectives byproviding a method for obtaining an optimal combination of sourceillumination and reticle mask features that are chosen such thatresulting image is optimized in accordance with a user-specified meritfunction and constraints.

According to a first aspect of the invention, a merit function is chosento describe a relationship between source parameters, reticle featuresor parameters, and desired image characteristics, and the merit functionis optimized subject to user-specified constraints on the resultingimage. Source parameters may include, for example, source direction andsource intensities. Reticle parameters may be defined, for example, interms of diffracted amplitudes. Image constraints may include, forexample, a predefined intensity at desired image feature edge points,and thresholds defining bright and dark areas of the image. The meritfunction may include, for example, the gradient of the imageperpendicular to the feature edges.

According to another aspect of the present invention, the features of areticle using a desired mask material, such as phase-shifting chrome,are formed based on a set of optimized diffracted amplitudes incombination with an optimized set of source parameters.

According to another aspect of the present invention, the transmittingor diffracting features of a reticle are formed such that dark areas ofthe desired image pattern are produced by destructive interference ofthe diffracted energy. Such a mask may be formed, for example, using aphase-shifting material such as phase-shifting chrome.

Also, according to another aspect of the present invention, alithographic system is provided that incorporates an optimizedcombination of source and reticle features, obtained using the method inaccordance with the present invention, in order to print a desiredpattern.

Also, according to a further aspect of the present invention, a computerprogram is provided that performs the method of obtaining a combinationof source parameters and reticle features such that characteristics of adesired image are optimized in accordance with a merit function.

According to the present invention, a method is described for printingan integrated circuit pattern on a semiconductor wafer having aphotoactive material thereon, the method comprising the steps of:

providing a desired wafer feature pattern having at least one waferfeature element;

deriving a merit function describing a relationship between anillumination source, a reticle, and an image, said source having atleast one source parameter, said reticle having at least one diffractivefeature, and said image having at least one image intensity;

selecting at least one constraint in relation to said desired waferfeature pattern that said at least one image intensity must satisfy;

selecting a combination of said at least one source parameter and saidat least one diffractive feature so that said merit function isoptimized in accordance with said at least one constraint;

illuminating said reticle with illumination energy from saidillumination source, so that said illumination energy is diffracted bysaid reticle and projected through a lens aperture to form said at leastone image intensity on the wafer;

exposing the photoactive material to said at least one image intensity;and

developing said exposed photoactive material to form a printed feature,

so that said printed feature conforms with said at least one waferfeature element of said desired wafer feature pattern in accordance withsaid constraints.

Also, according to the present invention, a method is described forselecting a combination of source illumination parameters anddiffraction mask features for projecting energy through a lens apertureto form an image pattern on a wafer, the method comprising the steps of:

providing a desired wafer feature pattern having at least one waferfeature element;

deriving a merit function describing a relationship between anillumination source, a reticle, and the image pattern, said sourcehaving at least one source source parameter, said reticle having atleast one diffractive feature, and said image pattern having at leastone image intensity;

selecting at least one constraint in relation to said desired waferfeature pattern that said at least one image intensity must satisfy; and

selecting a combination of said at least one source parameter and saidat least one diffractive feature,

so that said merit function is optimized in accordance with said atleast one constraint.

A computer program product according to the present invention isdescribed for selecting a combination of source illumination parametersand diffraction mask features for projecting energy through a lensaperture to form a desired image, the computer program productcomprising computer readable instructions for performing a method havingthe following steps:

causing a computer to store a desired wafer pattern having at least onewafer feature element;

causing the computer to compute a merit function describing arelationship between an illumination source, a reticle, and an imagepattern, said source having at least one source source parameter, saidreticle having at least one diffractive feature, and said image patternhaving at least one image intensity;

storing at least one constraint in relation to said desired waferfeature pattern that said at least one image intensity must satisfy; and

selecting a combination of said at least one source parameter and saidat least one diffractive feature,

so that said merit function is optimized in accordance with said atleast one constraint.

Also described, according to the present invention, is a machinereadable storage medium having stored therein a program of instructionsexecutable by the machine to perform method steps for selecting acombination of source illumination parameters and diffraction maskfeatures for projecting energy through a lens aperture to form a desiredimage, said method steps comprising:

storing a desired wafer feature pattern having at least one waferfeature element;

storing instructions for causing a computer to compute a merit functiondescribing a relationship between an illumination source, a reticle, andan image pattern, said source having at least one source sourceparameter, said reticle having at least one diffractive feature, andsaid image pattern having at least one image intensity;

storing at least one constraint in relation to said desired featurepattern that said at least one image intensity must satisfy; and

selecting a combination of said at least one source parameter and saidat least one diffractive feature,

so that said merit function is optimized in accordance with said atleast one constraint.

According to the present invention, a lithographic system is describedfor printing a desired wafer feature pattern on a semiconductor waferincluding a photoactive material, the system comprising:

an illumination source having at least one source parameter;

a reticle having at least one diffractive feature; and

a lens;

said illumination source, said reticle and said lens being arranged sothat said source illuminates said reticle so as to produce a pluralityof diffracted amplitudes and said plurality of diffracted amplitudes arecollected by said lens and projected to form a image on thesemiconductor wafer, the image having at least one image intensity andwherein

said at least one source parameter and said at least one diffractivefeature are selected in accordance with a merit function describing arelationship between said at least one source parameter, said pluralityof diffracted amplitudes, and said at least one image intensity andwherein said merit function is optimized in accordance with at least oneconstraint that said at least one image intensity must satisfy,

so that exposing the photoactive material to said at least one imageintensity and developing said exposed photoactive material forms atleast one printed feature that substantially conforms with the desiredwafer feature pattern.

A reticle according to the present invention is described fordiffracting illumination energy to form a desired image pattern having apattern of intensities, the desired image pattern having a bright areain which the intensities within the bright area exceed a predeterminedbright threshold and having a dark area in which the intensities withinthe dark area are less than a predetermined dark threshold, the reticlecomprising a pattern of phase-shifting material arranged so that thedark area is formed by destructive interference of diffractedillumination energy. A reticle according to the present invention isalso described wherein said phase-shifting material comprisesphase-shifting chrome material.

The novel features believed to be characteristic of this invention areset forth in the appended claims. The invention itself, however, as wellas other objects and advantages thereof, may be best understood byreference to the following detailed description of an illustratedpreferred embodiment to be read in conjunction with the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates key elements of a typical lithographic system asknown in the art.

FIG. 1A illustrates the relationship between the electric fieldamplitude of a diffracted wave front and the intensity at the imageplane as known in the art.

FIG. 2 illustrates the unit cell of a desired image pattern for oneembodiment of the present invention.

FIG. 3 illustrates collected orders from on-axis illumination for theunit cell of FIG. 2.

FIG. 3A illustrates collected orders from off-axis illumination for theunit cell of FIG. 2.

FIG. 4 illustrates a desired wafer feature pattern, for example, at anisolation level.

FIG. 4A illustrates a prior art mask using biased shapes to compensatefor line-shortening.

FIG. 5 illustrates a flow chart describing an embodiment of the methodfor printing a desired wafer pattern in accordance with the presentinvention.

FIG. 5A illustrates a flow chart describing an embodiment of thesolution of a simplified merit function in accordance with the presentinvention.

FIG. 6 illustrates a desired feature pattern for one embodiment of theinvention, typical for an isolation level.

FIG. 7 illustrates diffracted orders or source directions correspondingto the pattern of FIG. 6, plotted in direction space.

FIG. 8 plots example cross-sections of desired image intensitiesrequired to produce the pattern of FIG. 6.

FIG. 9 illustrates an optimized source pattern obtained in accordancewith the method of the present invention for the desired pattern of FIG.6.

FIG. 10 illustrates an optimized mask pattern using chromeless masktechnology obtained in combination with the optimized source of FIG. 9,using the method in accordance with the present invention for thedesired pattern of FIG. 6.

FIG. 11 illustrates a chromeless mask derived from the optimized patternof FIG. 10, using superimposed rectangular shapes.

FIG. 12 illustrates intensity contours for an image obtained using theoptimized source of FIG. 9 in combination with the mask of FIG. 11.

FIG. 13 illustrates a phase-shifting chrome mask using superimposedrectangular shapes derived from an optimized phase-shifting chromepattern obtained in combination with the optimized source of FIG. 9 forthe desired pattern of FIG. 6.

FIG. 14 illustrates a desired wafer feature pattern, as typical for aDRAM capacitor level.

FIG. 15 illustrates the optimized source obtained using the method inaccordance with the present invention for the pattern of FIG. 14.

FIG. 16 illustrates the optimized mask using chromeless mask technologyobtained in combination with the optimized source of FIG. 15 inaccordance with the present invention.

FIG. 17 illustrates an approximation of a chromeless mask derived fromthe optimized mask of FIG. 16 using superimposed rectangular shapes.

FIG. 18 illustrates intensity contours of the image obtained using thecombination of the optimized source of FIG. 15 and the mask of FIG. 17,including superimposed shapes from the mask of FIG. 17 and the desiredpattern from FIG. 14.

FIG. 19 illustrates the process window obtained using the optimizedsource of FIG. 15 and mask of FIG. 17 in accordance with the presentinvention, compared to process windows obtained using prior art sourceand mask combinations.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In accordance with the present invention, a method is described foroptimally selecting reticles and source illumination for use in opticalprojection systems, for example, imaging systems of the type used inoptical lithography, and a projection imaging system that operates inaccordance with such a method.

The present invention exploits the fact that the amplitudes of a wavefront diffracted by a reticle will be dependent on both illuminationamplitude and the diffractive properties of the mask.

Consider again, for example, a desired feature pattern illustrated inFIG. 4 which is represented by a unit cell as in FIG. 2, a horizontalrepeat dimension 203, and a staggered pitch indicated by the diagonalrepeat dimension 205 (alternatively indicated by the vertical pitch201). As previously discussed, using this unit cell as a repeatedpattern in a diffraction grating and illuminated by an on-axis beamresults in diffracted orders which can be plotted in direction space asindicated in FIG. 3. For example, the +2 order represented by thehorizontal repeat distance 203 is represented by the point 301 and the−2 order is represented by the point 310. Similarly, points 305 and 319represent the +2 and −2 orders based on the vertical repeat distance201. Other orders are diffracted both horizontally and vertically, suchas order 308, denoted as the {−1,+1} order. For reference, the numericalaperture (NA) 350 of the lens is also plotted. The only orders collectedby the lens are 300, 301, 310, 303, 308, 313, and 312.

It is possible to design a projection system having a combination ofillumination amplitudes and mask features such that different orders ofthe diffracted wave front collected by the lens aperture 350 haveamplitudes that are independently adjustable. For example, the amplitudeof the collected order 301 can be adjusted independently of order 303.

However, not all of the orders collected can be adjusted independentlybecause of the symmetry of the repeat dimensions. In this example,orders 301 and 310 are constrained to have the same amplitudes. In thisexample, as in many lithographic patterns, the pattern is bilaterallysymmetric about the horizontal and vertical axes, in which case onequadrant of the diffraction pattern essentially determines the otherthree. Thus, orders 303, 308, 312 and 313 must have the same amplitudesdue to the symmetry of the pattern. Therefore, there are really only 3independent orders collected in the example shown in FIG. 3. Moreover,to avoid strong focus sensitivity it is necessary to restrict the phaseof the reticle transmittance to 0° or 180°, making the mask amplitudetransmittance a purely real quantity, i.e. non-complex. The real part ofthe mask diffraction pattern therefore has radially even symmetry andthe imaginary part odd symmetry. This means, for example, that order 303cannot be adjusted independently of order 313, regardless of whether ornot the mask patterns are bilaterally symmetric. Other order pairs atequal and opposite radii, such as 308 and 312 or 301 and 310, areconstrained in the same way.

On the other hand, if the illumination comes from the 301 direction,then the orders collected will be shifted as shown in FIG. 3A. In thiscase, there are five independent orders collected, 300, 301, 302, (303and 312), and (304 and 311), providing an increase in the number oforders that can be independently adjusted. Thus, if illumination comesfrom the 301 direction, a large number of orders can be independentlyadjusted. By properly adjusting the illumination direction and thecollected orders together, the number of adjustable degrees of freedomin the image can be increased, and improved performance can be achievedover what is known in the art.

One embodiment of a method according to the present invention isillustrated in FIG. 5. The method achieves enhancement of the image withrespect to a desired IC pattern by simultaneously choosing parameters ofsource illumination and mask transmission features in a lithographicimaging system so that features of the projected image are optimized asmeasured by a merit function in accordance with a set of constraints onthe image features and lithographic process parameters. The first step500 consists of providing an desired image pattern, for example, apattern in which CD is very close to the resolution limits of thelithography system. As an example, for this embodiment, the desiredpattern to be printed is illustrated in FIG. 6 (essentially the same asthe pattern in FIG. 4), having rectangular features of width 401 equalto the basic dimensional unit of the cell F, and a vertical spacing 402between the rectangles also equal to F. The length 405 of therectangular features is equal to 6.5 F. The desired horizontal spacing408 between the tips of the rectangles is 1.5 F. The next step 510 ofthe method in accordance with the present invention is to derive a meritfunction that will be optimized when an appropriate set of source andmask parameters are selected. Considerations in deriving a meritfunction in this embodiment are discussed in the following paragraphs.

As noted above, to avoid strong focus sensitivity it is desirable thatthe mask amplitude transmittance be pure real, i.e. the phase of thetransmittance should be restricted to 0° or 180°. This means that thereal part of the electric field diffracted by the mask will be radiallyeven and the imaginary part will be radially odd. Moreover, lithographicpatterns are often bilaterally symmetric about horizontal and verticalaxes, in which case one quadrant of the diffraction pattern essentiallydetermines the other three. When the k-factor is sufficiently large,this symmetrical diffraction pattern is simply collected by the lens,and then transferred to the wafer, thereby providing the appropriatesymmetry in the image. However, it is not necessary to impose a symmetryconstraint on individual collected diffraction orders themselves becausethe required symmetry can be achieved by using an appropriately selectedsymmetric source illumination distribution.

A merit function based on the image intensity may be defined in terms ofunknown values of source directions and intensities and diffractedwavefront amplitudes expressed as diffraction order amplitudes usingstandard equations of image formation which are based on well knownprinciples of optics. In this embodiment, a merit function is chosenthat describes the gradient of the image across the edges of features atselected critical positions, such that the smallest slope at allselected positions is maximized. Such a merit function will ensure goodimage fidelity. But many other merit functions may also be chosen inaccordance with this invention.

A set of constraints are required in order to solve an optimizationproblem. In this example (block 530), constraints on the projected imageare imposed, for example, that critical edge positions share a commonintensity, Q, to eliminate CD errors and prevent line shortening.Referring to FIG. 6, several points along the edge of the rectangularfeatures (604, 606, and 608) may be selected, as well as points at thetips of the features (points 610 and 612) at which this constraint mustbe satisfied. Other constraints chosen require that the ratio of themaximum intensity generated in dark shapes to the minimum intensity inbright areas be below a development threshold, to ensure proper imagetopology. Points (such as locations 601, 602, 603, 614, 616, and 618)may be selected at which those constraints on intensity ratios must besatisfied. Additional constraints are imposed (block 540) to: i) enforcegeometric restrictions on the size of the source regions; ii) requireachievement of minimum acceptable pupil fill; iii) require adequateexposure in bright areas; and iv) prevent unwanted exposure in darkareas. The method in accordance with the present invention is notrestricted to these constraints, but could include alternate oradditional constraints, for example, by taking aberrations into accountby approximating each source region, or subdivision of a region, as alocally centered point, or as a uniformly spaced collection of points.Using such an approximation, sample points can include points inmultiple focal planes, and under multiple aberration conditions.

The merit function can then be maximized (block 550 in FIG. 5) to solvefor the unknown source directions and intensities and diffraction orderamplitudes using standard techniques for global optimization that areknown in the art. The resulting optimized source directions, sourceintensities, and diffracted wave front (or diffraction order) amplitudesare then implemented within a lithographic system. The reticle maskfeatures corresponding to the derived optimal diffraction orderamplitudes may be readily determined since the relationship betweendiffraction pattern and reticle transmittance is linear, based on aFourier transform. User-defined source shapes may be implemented using anumber of techniques known in the art, ranging from simple apertures inthe entrance pupil to customized diffractive elements.

It is known in the art that for a general merit function no fully globalalgorithm can be guaranteed to perform better than simple exhaustivegrid search of the parameter space. However, in determining theoptimized off-axis illumination directions and diffracted wavefronts,this embodiment of the present invention exploits the particularstructure of the optimization problem to find a useful solution on a farmore rapid basis.

The difficulty in the present problem is that the merit function is notconcave, which is known to imply the presence of multiple local maxima;indeed, the plane-wave orders that comprise the image are intrinsicallyoscillatory, giving rise to a great many local maxima in the meritfunction. As discussed above, an optimized solution to the meritfunction of this problem cannot be readily obtained with goodperformance from conventional starting solutions, which are based on thetarget wafer patterns. It is therefore important that the optimizationtechnique achieve good global performance; robust local convergence isinsufficient. To achieve global convergence (block 550) in an efficientway this embodiment of the present invention utilizes a two-partstrategy illustrated in FIG. 5A to solve for an optimized wave front(resulting in a set of illumination and diffraction order amplitudesdefined in the pupil plane), followed by the determination of thereticle pattern based on the optimized wave front (block 553):

1) Calculate the global optimal solution (block 551) for a simplifiedversion of the merit function (block 510) and constraints (blocks 530and 540).

2) Use a local optimization technique (block 552) to refine thesimplified global solution of step 1 against more complete criteria.

3) Calculate a reticle pattern (included in block 553) that provides theoptimized wave front determined in step 2. As long as step 1 (block 551)provides a good starting solution, the robustness of widely availablelocal optimization routines allows the incorporation of detailedoptimality criteria in step 2 (block 552). For example, the image modelused with the step 2 local optimizer can incorporate a finely spacedillumination grid with 0.02 NA steps, and a focal sampling of 0.3λ/NA².An example of a commercially available local optimizer that can be usedto obtain the step 2 solutions is the FindMinimum optimizer provided byWolfram Research, Inc. in their Mathematica™ package, which is based onBrent's algorithm. Thus, the solution of step 2 will be fairlystraightforward.

The calculation of the optimized reticle shapes, which is included instep 3 (block 553), can also be performed using a relatively simpleapproach, which is described more fully below, that exploits thelinearity of the Fourier transform process that governs diffraction.

In this embodiment, the global optimization of step 1 (block 551) isperformed using a simplified version of the problem that considers onlyan aberration-free image, i.e. considerations of defocus and the fixedaberrations are deferred to step 2, and in step 1 are approximated aszero. Alternatively, aberrations and defocus can be taken into accountin an approximate way during the step 1 optimization. When the targetpatterns are periodic, or may have periodic boundary conditions appliedthereto, the aberration-free approximation allows a partitioning of thecontinuous space of possible source illumination directions into afairly small number of distinct regions or zones, since two directionsare equivalent (when aberrations are zero) if they direct the same setof diffraction orders into the collection pupil. FIG. 7 is a diagramsimilar to FIG. 3 in direction space of light diffracted into the pupilfrom a reticle projecting the desired pattern illustrated in FIG. 6.Alternatively, the diagram in FIG. 7 can also be viewed as representingthe illumination directions incident on the reticle. The range ofillumination directions for which a given order is collected by the lenspupil can be represented by circles centered on the point correspondingto the order collected from an on-axis illumination beam. For example,the interior of the circle 703′ represents the range of illuminationdirections that provide collection of order 703. Thus, the interior ofeach of the regions formed by the intersections of the circlesrepresents a range of directions that provide the same set of collectedorders, and therefore the same focused image. If aberrations areneglected, the illumination can be fully represented without furtherloss of generality using pupil regions from separate orders that overlapwithin one quadrant, since illumination outside of a quadrant can beobtained by mirroring. In FIG. 7, the circles indicating the range ofcollectable directions for each order that overlap in the upper righthand quadrant are plotted. In this embodiment, variables representingthe source intensity from different source directions are selected fromoverlapping pupil regions from the area circumscribed by the firstquadrant (indicated by points 792 to 700 to 791) enclosed by the smallercircle 250 representing approximately 85% of NA. Within this firstquadrant, there are 41 distinct areas having overlapping pupil regions.Thus, 41 distinct illumination variables were selected representing theilluminating intensity from each of the 41 different pupil regions shownin FIG. 7. For example, one variable is chosen to represent the region777 in which only orders 700, 701, 702, 703, 704, 711, and 712 arecollected. These unknown source directions are denoted as a vectorvariable {right arrow over (s)} (of length 41 in this example). Notethat each element of {right arrow over (s)} represents a set of 1, 2, or4 equally intense illuminating directions that impinge on the mask frommirrored directions.

Usually the illuminator will fill all open illumination directions witha fixed power per unit solid angle. In this case, one of the constraintson the solution (block 540) is that the source intensity from eachsource direction j must satisfy:

0≦s _(j) ≦S _(Max,j)

where s_(Max,j) is the area of the jth illumination region in the pupil.If the source distribution is defined by diffractive source elements itis more appropriate to constrain the summed intensity.

The m,nth diffraction order of an idealized wavefront is ordinarilydefined in the prior art as the amplitude a_(m,n) that diffracts fromthe reticle in a direction d=(mλ/p_(x),nλ/p_(y)), in which p_(x) andp_(y) are the unit cell periodicities. As noted above, the individualcollected orders are not all independent. However, in accordance withthis invention it is desirable that the unknown diffracted amplitudevariables represent independently adjustable components of thewavefront. Therefore, in accordance with the present invention, m and nare treated as non-negative and a_(m,n) then represents a singlenon-redundant unknown. For a given source direction j, the amplitude atthe wafer image plane b_(m,n,j) that is produced by an unknowndiffracted amplitude a_(m,n) can therefore include the result ofinterference between superimposed waves from the ±m, ±n directions. Inother words, the image amplitude b_(m,n,j) may be given by$\begin{matrix}{^{2{\pi {({\frac{mx}{p_{x}} + \frac{ny}{p_{y}}})}}},{^{2\pi \quad i\quad \frac{mx}{p_{x}}}{\cos \left( \frac{ny}{p_{y}} \right)}},{^{2\pi \quad i\frac{ny}{p_{y}}}{\cos \left( \frac{mx}{p_{x}} \right)}},{{or}\quad {\cos \left( {\frac{mx}{p_{x}} + \frac{ny}{p_{y}}} \right)}},} & \lbrack 2\rbrack\end{matrix}$

depending on whether or not the associated negative orders in the x,ymirror directions are simultaneously captured in the pupil forillumination direction j. It is convenient to write the diffractedamplitudes a_(m,n) and the wafer image amplitudes b_(m,n,j) as vectors;in other words, {right arrow over (a)} for the unknown diffracted orderamplitudes, representing all orders that can be captured from anyfeasible illumination direction, and {right arrow over (c)}₁ and {rightarrow over (c)}₂ for the real and imaginary parts, respectively, of thewafer image amplitude {right arrow over (b)}. Also, in this embodiment,it is possible to simplify the notation, since in this case the targetpatterns are symmetric. If the reticle were illuminated by a coherentoff-axis beam (i.e. a highly non-symmetric illumination with nomirroring), the intensity produced at a point (x,y) on the wafer wouldbe given by $\begin{matrix}{{{I_{Asym}\left( {x,y} \right)} = {\sum\limits_{j = 1}^{J_{Max}}\quad {\sum\limits_{h = 1}^{2}\quad {s_{j}\left( {{\overset{\rightarrow}{c}}_{j,h} \cdot \overset{\rightarrow}{a}} \right)}^{2}}}},} & \lbrack 3\rbrack\end{matrix}$

where an inner summation over index h (where h goes from 1 to 2) isincluded to separate real and imaginary parts. In this embodiment,J_(Max) is equal to 41, representing the illumination directionsselected from the first quadrant of the direction space diagram in FIG.7. Thus, in order to provide proper symmetry in the image, the reticlemust be illuminated symmetrically from mirrored directions, so that thetotal wafer-plane intensity for a symmetrical object becomes

I(x,y)=I _(Asym)(x,y)+I _(Asym)(−x,y)+I _(Asym)(x,−y)+I_(Asym)(−x,−y).  [4]

Equation [4] can be simplified by adding an additional index q (rangingfrom 1 to 4) to {right arrow over (c)} to distinguish the four mirroredillumination directions: $\begin{matrix}{{{I\left( {x,y} \right)} = {\sum\limits_{q = 1}^{4}\quad {\sum\limits_{j = 1}^{J_{Max}}\quad {\sum\limits_{h = 1}^{2}\quad {s_{j}\left( {{\overset{\rightarrow}{c}}_{q,j,h} \cdot \overset{\rightarrow}{a}} \right)}^{2}}}}},} & \lbrack 5\rbrack\end{matrix}$

where care must be taken to properly treat non-diagonal reticle andillumination orders.

The simplified global optimization step 1 (block 551 indicated in FIG.5) of this embodiment can now be represented as a generalized fractionalprogram:${{Maximize}\quad \Psi \quad \left( {\overset{\rightarrow}{s},\overset{\rightarrow}{a}} \right)} \equiv {\underset{r}{Min}\left( \frac{\sum\limits_{q = 1}^{4}\quad {\sum\limits_{j = 1}^{J_{Max}}\quad {\sum\limits_{h = 1}^{2}\quad {{s_{j}\left( {{\overset{\rightarrow}{c}}_{q,j,h,r} \cdot \overset{\rightarrow}{a}} \right)}\left( {\nabla{\bot{{\overset{\rightarrow}{c}}_{q,j,h,r} \cdot \overset{\rightarrow}{a}}}} \right)}}}}{\sum\limits_{q = 1}^{4}\quad {\sum\limits_{j = 1}^{J_{Max}}\quad {\sum\limits_{h = 1}^{2}\quad {{s_{j}\left( {{\overset{\rightarrow}{c}}_{q,j,h,r} \cdot \overset{\rightarrow}{a}} \right)}}^{2}}}} \right)}$

subject to: $\begin{matrix}{\quad \begin{matrix}{{0 \leq \overset{\rightarrow}{s} \leq S_{{Max},j}},{{\sum\limits_{j = 1}^{J_{Max}}\quad s_{j}} \leq S_{Min}},} & \quad \\{{{\sum\limits_{q = 1}^{4}\quad {\sum\limits_{j = 1}^{J_{Max}}\quad {\sum\limits_{h = 1}^{2}\quad {s_{j}\left( {{\overset{\rightarrow}{c}}_{q,j,h,r} \cdot \overset{\rightarrow}{a}} \right)}^{2}}}} = Q},} & \left( {\left. {\forall{r{{1 \leq r \leq r_{Max}}}}} \right),} \right. \\{{\sum\limits_{q = 1}^{4}\quad {\sum\limits_{j = 1}^{J_{Max}}\quad {\sum\limits_{h = 1}^{2}\quad {s_{j}\left( {{\overset{\rightarrow}{c}}_{q,j,h,u} \cdot \overset{\rightarrow}{a}} \right)}^{2}}}} \geq {I_{Bright}Q}} & \left( {\left. {\forall{u{{1 \leq u \leq u_{Max}}}}} \right),} \right. \\{{\sum\limits_{q = 1}^{4}\quad {\sum\limits_{j = 1}^{J_{Max}}\quad {\sum\limits_{h = 1}^{2}\quad {s_{j}\left( {{\overset{\rightarrow}{c}}_{q,j,h,v} \cdot \overset{\rightarrow}{a}} \right)}^{2}}}} \leq {I_{Dark}Q}} & {\left. \left( {\forall{v{{1 \leq v \leq v_{Max}}}}} \right. \right).}\end{matrix}\quad} & \lbrack 6\rbrack\end{matrix}$

Here the index r runs over sample points (x_(r),y_(r)) along the edgesof the target patterns, for example, as indicated by points 604, 606,608, 610, and 612 in FIG. 6 and in FIG. 12. ⊥{right arrow over (c)}represents the derivative of c in a direction normal to the featureedge. Ψ is a merit function that represents the worst-case log-slopearising at feature edges in the image. Optimization of Ψ ensures thatthe shallowest slope among feature edges is as steep as possible. Ifdesired, a weighting factor can be applied to the log-slope at eachposition, to reflect differing CD tolerances. The indices u and v runover sample points in image regions that must be bright (for example,points 614, 616 and 618 indicated in FIG. 6) and dark (for example,points 601, 602 and 603 indicated in FIG. 6), respectively.

Constraints are imposed (blocks 530 and 540) to: i) enforce geometricrestrictions (block 540) on the size of the s_(j) source regions, asindicated by S_(Max, j); ii) require achievement of minimum acceptablepupil fill (block 540), as indicated by S_(Min); iii) prevent lineshortening and other CD errors in the printed pattern by forcing alledges (block 530) (for example, at points 604, 606, 608, 610 and 612indicated in FIG. 6, FIG. 12, and FIG. 8) to print at a common (but notpre-specified) intensity Q 810 (as indicated in FIG. 8) (block 530); iv)require adequate exposure in bright areas (block 530) (for example atpoints 614, 616, and 618 in FIG. 6), as indicated by the termI_(Bright)Q; and v) prevent unwanted exposure in dark areas (block 530)(for example, at points 601, 602, and 603 in FIG. 6), as indicated bythe term I_(Dark)Q. The equation [6] approximation restricts thedistinguishable source variables to the discrete set of pupil zonesdefined by an aberration-free image (as in FIG. 7). However, it ispossible to take aberrations into account if each source region isapproximated as a locally centered point, or as a uniformly spacedcollection of points within the region. This alternate approximation isused in this embodiment of the present invention, and sample points inmultiple focal planes are included, and under multiple aberrationconditions.

Techniques are known in the art for solving fractional optimizationproblems like equation [6], typically reducing them to a parametricproblem in the difference between numerator and denominator. Equation[6] can also be approximated as a cubic polynomial optimization, andsolved, for example by a homotopy method (illustrated below for thequadratic case). Nonetheless, equation [6] is still a fairly difficultnonlinear problem.

In this embodiment of the present invention, an approximate solutionscheme for equation [6] is used which exploits the fact that twosimplified variants of equation [6] are more readily solvable. First, asexplained further below, if the diffracted wavefront orders a are fixed,it is possible to find the globally optimum solution for the sourceintensities {right arrow over (s)}. Second, if illuminating light isincident from only a single direction (more precisely, a single group ofmirrored directions), equation [6] reduces to a nonconvex quadraticoptimization problem, which can be solved more easily than the generalcase (see below).

In addition, one further approximation is made in this embodiment of thepresent invention. In all present steppers, illumination is restrictedto directions that are within the NA. The various sets of collectedorders associated with each of the different illumination zones are thenhighly non-disjoint; in fact, each consists of a different collectiondrawn from the same limited set of diffraction orders. The waferinterference patterns produced by different source directions mayinclude varying numbers of mirrored copies of a given order;nonetheless, only a limited number of independent amplitude orders arecollected even from a full illumination pupil. This means that the setsof collected orders produced by different illumination beams, thoughfree from interference with one another, are not independentlyoptimizable (although each can be weighted independently by adjustingthe corresponding component of {right arrow over (s)}.) For example, inthis embodiment, there are 41 independent illumination regions for thein-focus image, but only 8 independent orders, namely 700, 701, 702,703, 704, 705, 706, and 716 (see FIG. 7). Note that the circleindicating the range of the 716 order only slightly intersects with the85% NA circle 250, so that its contribution can be ignored in thisexample without significantly changing the solution. Suppose then thatthe interference pattern from a particular direction j₁ simultaneouslyprovides an image that is not only suitably dark at all desired(x_(v),y_(v)) dark image points (for example, 601, 602, and 603 in FIG.6), but is also dark at some subset u_(1d) of (x_(u),y_(u)) desiredbright image points (for example, at some subset of the points 614, 616,and 618 in FIG. 6), while at the same time being bright at the remainingu_(1b) (x_(u),y_(u)) desired bright image points (for example, at theremaining points amongst 614, 616, and 618). Given the strong overlapbetween the different collected order sets, it can be assumed that it isunlikely that the interfering orders from another illumination region j₂will simultaneously be bright at the subset u_(1d) of the (x_(u),y_(u))desired image bright points at which the first set of orders is dark, inaddition to being dark at all (x_(v),y_(v)) dark points, and at the sametime dark at some of the desired bright image points at which the firstset was dark. Such a situation would require that the same limited setof orders simultaneously satisfy constraints involving two illuminationdirections, instead of just one (though the doubled constraints are notuniquely determined). Since such combinations are unlikely, in thisembodiment it is assumed as an approximation that the optimum solutionwill not comprise such a system. Accordingly, only those illuminationdirections that individually satisfy all constraints for bright and darkregions are considered, leading to the following iterative method forsolving optimization problem 550 in FIG. 5 for the reticle andillumination source:

Step 0) Initial Characterization:

a. For each of the J_(Max) source directions (ranging from 1 through 41in this embodiment), calculate a solution for diffracted mask amplitudes{right arrow over (a)} that is globally optimum under simplifiedcriteria, for example, ignoring the edge constraints and minimum pupilfill (i.e. initializing S_(Min) to 0).

b. Initialize the diffracted mask amplitudes {right arrow over (a)} tothe best value obtained in step 0.a.

Step 1) Global Optimum of Simplified Merit Function (block 551):

a. Calculate the globally optimum source distribution {right arrow over(s)} given the current values of {right arrow over (a)} and s_(Min).

b. Use a local algorithm to optimize {right arrow over (s)} and {rightarrow over (a)} together including the constraints at selected imagepoints at multiple focal planes (e.g. edge constraints, ratios of brightto dark points) (block 530).

c. Increase S_(Min) by a small increment (e.g. by 5% of pupil area) andreturn to step 1.a, repeating until pupil is filled (block 540) orconstraints (block 530) cannot be met.

Step 2) Local Optimization to Refine Global Solution (block 552):

a. Fix S_(Min) at desired final level and choose corresponding solutionfor {right arrow over (s)} and {right arrow over (a)} from step 1.

b. Refine the solutions for {right arrow over (s)} and {right arrow over(a)} using a local optimization technique with more complex constraints(e.g. high-NA, thin-film and resist effects). Alternatively, some ofthese more complex constraint criteria can also be applied in step 1.

Step 3) Calculate the optimum reticle pattern (block 553) that providesthe diffracted wavefront {right arrow over (a)} obtained in step 2.

These steps are now considered in more detail.

In calculating the step 0.a amplitude sets {right arrow over (a)}_(j),the constraints on equal feature bias and minimum pupil fill weredeferred to step 1. As a further simplification, the optimization instep 0 can be performed against the finite difference between adjacentdark and bright points across feature edges (deferring optimizationagainst true log-slope until step 1. Moreover, the overall intensityscaling of the amplitudes {right arrow over (a)} is arbitrary until thestep 3 mask calculation. Thus, in this embodiment, the intensity at edgepoints such as 640, 606, 608, 610, and 612 is constrained to be greaterthan or equal to 1, which means that the log-slope will be maximized ina finite difference sense if the intensity at dark points on theopposite side of the edges is minimized. The step 0.a optimizationproblem for the jth source direction is then: $\begin{matrix}{{\Phi_{j}\left( \overset{\rightarrow}{a} \right)} \equiv {\sum\limits_{r = 1}^{R_{Max}}\quad {\sum\limits_{q = 1}^{4}\quad {\sum\limits_{h = 1}^{2}\quad \left( {{\overset{\rightarrow}{c}}_{q,j,h,r} \cdot \overset{\rightarrow}{a}} \right)^{2}}}}} & \lbrack 7\rbrack\end{matrix}$

subject to: $\begin{matrix}{{\sum\limits_{q = 1}^{4}\quad {\sum\limits_{h = 1}^{2}\quad \left( {{\overset{\rightarrow}{c}}_{q,j,h,u} \cdot \overset{\rightarrow}{a}} \right)^{2}}} \geq 1} & {\left. \left( {\forall{u{{1 \leq u \leq u_{Max}}}}} \right. \right).}\end{matrix}$

which can be re-written in matrix form as:

Minimize Φ_(j)({right arrow over (a)})={right arrow over (a)}^(T) A ₀{right arrow over (a)}

subject to:

{right arrow over (a)}^(T) A _(u) {right arrow over (a)}≧1 (∀u|1≦u≦u≦u_(Max)).  [8]

Even though matrices A₀, A₁, . . . A_(uMax) are positive definite,equation [8] is a nonconvex problem because the inequality constraintsare lower bounds.

Nonconvex quadratically constrained quadratic programming problems suchas equation [8] are currently an area of active research; they are notrapidly solvable in the general case. However, solutions can usually beobtained in practice for reasonable-sized problems. Moreover, we canexploit the special structure of equation [8] to obtain the globalsolution on a far more rapid basis than is possible in the general case.

The equation [8] ellipsoids share a common center. For the currentembodiment of step 0.a, the search space for the optimization process isorganized by using a spherical triangulation of the hypersphere whichbounds the ellipsoids, with a first set of nodes for thesehypertriangles (i.e. simplices) being defined by unit vectors along theprincipal axes of the ellipsoids. The other half of the node set is thengenerated by splitting these hypertriangles through the addition of aunit vector at the midpoint of each. For small to moderate-sizedproblems one can then refine the feasible solutions along each ray usinga local algorithm. Moreover, the ellipsoids for the initial searchvectors can be constructed in a space of greatly reduced dimensionality.This is because the eigenvalues of matrix A₀ in equation [8] must rangebetween very small and very large amplitudes, corresponding to the widerange of intensities that can be printed at feasible values of parameterρ in equation [1]. The eigenvalues of matrix A₀ correspond to differentaverage intensities over the dark sample points. It is convenient tosimultaneously diagonalize A₀ in terms of the average intensities atboth bright and dark sample points. The initial search hypersphere canthen be restricted in dimensionality to the subspace spanned by thesmallest eigenvalues for the dark points. Enough eigenvalues from thejoint diagonalization must be included such that at least oneeigenvector provides intensity greater than 1 for each of the brightsample points, where the value 1 is an arbitrary set point for minimumintensity at each bright sample point. Simultaneous diagonalizationagainst more than two sets of basis vectors is not generally possible,and the retained eigenvalues and eigenvectors are not sufficient inthemselves to account for interference between individual bright samplepoints. However, eigenvalue selection typically reduces thedimensionality of the search hypersphere (and therefore thedimensionality of the ellipsoids whose principal axes determine thesearch spherical triangles) by a factor of about 2, substantiallyreducing computation time. The local optimization along each ray shouldtake place in the full vector space, or at least in a space containing alarger proportion of the dark region eigenvectors.

A second embodiment of step 0.a uses a branch-and-bound algorithm asknown in the art. Here the search space is iteratively subdivided toremove regions which cannot contain the solution. Often the partitioningis based on interval arithmetic.

A third embodiment of step 0.a uses a homotopy algorithm as known in theart, in which the numerical bound for all inequality constraints exceptone (the uth) is changed from 1 to t, with t initialized to 0. The uthconstraint then becomes active. The Lagrangian at t=0 is therefore:

 L _(j)(t=0; μ,{right arrow over (a)})={right arrow over (a)} ^(T) A ₀{right arrow over (a)}−μ({right arrow over (a)}A _(U) {right arrow over(a)}−1).  [9]

Setting to zero the derivatives with respect to each component of {rightarrow over (a)} we obtain the following necessary condition for asolution:

|B|=0 where B=A ₀ −μA _(U).  [10]

The solution to equation [10] is the eigenvector of A₀ with minimumeigenvalue in a basis in which A_(U) is diagonalized and rescaled to theidentity matrix. Parameter t is then incremented in small steps with alocal optimizer applied at each step; finally equation [8] is solvedwhen t reaches 1.

A global optimum is in principal guaranteed if a homotopy algorithm isapplied to the full Lagrangian for equation [8]. Defining slackvariables {right arrow over (y)}² to represent the distance of thebright sample points from the constraint boundary (in intensity units),the Lagrangian for equation [8] becomes: $\begin{matrix}{{L_{j}\left( {\overset{\rightarrow}{y},\overset{\rightarrow}{\mu},\overset{\rightarrow}{a}} \right)} = {{{\overset{\rightarrow}{a}}^{T}A_{0}\overset{\rightarrow}{a}} + {\sum\limits_{u = 1}^{U_{Max}}\quad {{\mu_{u}\left( {{{\overset{\rightarrow}{a}}^{T}A_{u}\overset{\rightarrow}{a}} - y_{u}^{2} - 1} \right)}.}}}} & \lbrack 11\rbrack\end{matrix}$

Differentiating with respect to the variables {right arrow over(y)},{right arrow over (μ)}, and {right arrow over (a)} we obtain theoptimality conditions: $\begin{matrix}\begin{matrix}{{\left( {A_{0} + {\sum\limits_{u = 1}^{U_{Max}}\quad {\mu_{u}A_{u}}}} \right)\overset{\rightarrow}{a}} = 0} & \quad \\{{{{\overset{\rightarrow}{a}}^{T}A_{u}\overset{\rightarrow}{a}} - y_{u}^{2}} = 1} & \left. \left( {\forall{u{{1 \leq u \leq u_{Max}}}}} \right. \right) \\{{\mu_{u}y_{u}} = 0} & \left. \left( {\forall{u{{1 \leq u \leq u_{Max}}}}} \right. \right)\end{matrix} & \lbrack 12\rbrack\end{matrix}$

Equation [12] represents a set of simultaneous quadratic equations inthe variables {right arrow over (y)},{right arrow over (μ)} and {rightarrow over (a)}. Simultaneous polynomial equations can be solved byhomotopy, as known in the art. Solution speed is insensitive to thenumber of dark points; in fact a dense sampling can be advantageousbecause it prevents formation of an undersampled manifold containing aninfinite number of solutions (which may affect robustness of thehomotopy). Speed is improved when a sparse sampling of bright points isused. Since CD uniformity is only enforced in steps 1 and 2 of thepresent embodiment (blocks 551 and 552, bright sampling for the step 0Lagrangian need only be maintained at a level that conforms to thesampling theorem.

Upon completion of step 0, the present embodiment has only provided whatis essentially a coherent solution. The coherent image is notsymmetrical, but it becomes so under fourfold illumination from themirrored directions.

For the step 1.a global optimization of the source distribution,equation [6] is solved for {right arrow over (s)}, with {right arrowover (a)} given. Equation [6] can be transformed into a linear probleminvolving a new set of variables.

z ₀ ,z ₁ ,z ₂ ,z ₃ , . . . ≡z ₀ ,{right arrow over (z)},  [13]

in the linear problem:

Minimize z₀,

subject to: $\begin{matrix}{\quad \begin{matrix}{{z_{0} + {\overset{\rightarrow}{z} \cdot {\sum\limits_{q = 1}^{4}\quad {\sum\limits_{h = 1}^{2}\quad {\left( {{\overset{\rightarrow}{c}}_{q,j,h,r} \cdot \overset{\rightarrow}{a}} \right)\left( {{\nabla{\overset{\rightarrow}{c}}_{q,j,h,r}} \cdot \overset{\rightarrow}{a}} \right)}}}}} \geq 0} & \left( {\left. {\forall{r{{1 \leq r \leq r_{Max}}}}} \right),} \right. \\{{0 \leq {S_{Min}z_{j}} \leq {S_{{Max},j}{\sum\limits_{k = 1}^{J_{Max}}\quad z_{k}}}},} & \left( {\left. {\forall{j{{1 \leq j \leq J_{Max}}}}} \right),} \right. \\{{\overset{\rightarrow}{z} \cdot {\sum\limits_{q = 1}^{4}\quad {\sum\limits_{h = 1}^{2}\quad \left( {{\overset{\rightarrow}{c}}_{q,j,h,r} \cdot \overset{\rightarrow}{a}} \right)^{2}}}} = 1} & \left( {\left. {\forall{r{{1 \leq r \leq r_{Max}}}}} \right),} \right. \\{{\overset{\rightarrow}{z} \cdot {\sum\limits_{q = 1}^{4}\quad {\sum\limits_{h = 1}^{2}\quad \left( {{\overset{\rightarrow}{c}}_{q,j,h,u} \cdot \overset{\rightarrow}{a}} \right)^{2}}}} = I_{Bright}} & \left( {\left. {\forall{u{{1 \leq u \leq u_{Max}}}}} \right),} \right. \\{{\overset{\rightarrow}{z} \cdot {\sum\limits_{q = 1}^{4}\quad {\sum\limits_{h = 1}^{2}\quad \left( {{\overset{\rightarrow}{c}}_{q,j,h,v} \cdot \overset{\rightarrow}{a}} \right)^{2}}}} = I_{Dark}} & {\left. \left( {\forall{v{{1 \leq v \leq v_{Max}}}}} \right. \right).}\end{matrix}\quad} & \lbrack 14\rbrack\end{matrix}$

Software for solving linear programming problems is widely available;the examples here were calculated using the LinearProgramming routine inthe Mathematica™ package from Wolfram Research, Inc. After equation [14]is solved for z0 and z the step 1.a source intensities are given by:$\begin{matrix}{\overset{\rightarrow}{s} = {\frac{S_{Min}\overset{\rightarrow}{z}}{\sum\limits_{k = 1}^{J_{Max}}\quad z_{k}}.}} & \lbrack 15\rbrack\end{matrix}$

As described above, it is straightforward to carry out the step 2 localoptimization (block 552) on the basis of a forward model of the imageformation process.

Step 3 of the present embodiment (block 553) is the computation of thereticle which provides the optimized diffraction amplitudes {right arrowover (a)}. First, the set of reticle patterns are computed that providethe brightest possible image consistent with the step 2 solution for{right arrow over (a)}. Once a definite set of reticle patterns aredetermined, the layout must be refined using criteria that are wellknown to those skilled in the art. For example, the optimized patternsshould be rendered on the mask as polygons, preferably as a set ofrectangles. The rectangles can be fairly coarse, e.g. of dimension onlymodestly smaller than critical CDs, so long as their Fourier transformpreserves the relative diffraction order amplitudes of the step 2solution.

For the basic reticle calculation the Fourier diffraction integral isapproximated over the continuous mask transmission function T(x,y) as asummation over discrete sample points, which may, for example, bedefined on the grid of the mask writing tool. The 2D array oftransmission sample points can be unraveled into a 1D vector of unknowns{right arrow over (T)} indexed by q: $\begin{matrix}\begin{matrix}{{\int_{{- p_{x}}/2}^{p_{x}/2}{\int_{{- p_{y}}/2}^{p_{y}/2}\quad {{x}\quad {y}\quad {T\left( {x,y} \right)}^{2\pi \quad {i{({\frac{mx}{p_{x}} + \frac{ny}{p_{y}}})}}}}}} \cong \quad {\sum\limits_{k = 1}^{K}{\sum\limits_{l = 1}^{L}\quad {{T\left( {x_{k},y_{l}} \right)}^{2\pi \quad {i{({\frac{{mx}_{k}}{p_{x}} + \frac{{ny}_{l}}{p_{y}}})}}}}}}} \\{\equiv \quad {\sum\limits_{g = 1}^{KL}\quad {T_{g}b_{g,m,n}^{\prime}}}} \\{\equiv \quad {\sum\limits_{g = 1}^{KL}\quad {T_{g}{b_{g,w}^{\prime}.}}}}\end{matrix} & \lbrack 16\rbrack\end{matrix}$

In equation [16] the symbol b′ has been introduced as shorthand for theexponential. Also, in the last line, the index w represents theparticular values of the x,y order indices m,n that correspond to thewth captured order in the set {right arrow over (a)}.

Step 3 now becomes a linear programming problem:

Minimize${{- {{Sign}\left\lbrack {\sum\limits_{w = 1}^{W_{Max}}\quad a_{w}} \right\rbrack}}{\sum\limits_{g = 1}^{KL}\quad {\sum\limits_{w = 1}^{W_{Max}}\quad {T_{g}b_{g,w}^{\prime}}}}},$

subject to: $\begin{matrix}{\quad {\begin{matrix}{{\sum\limits_{g = 1}^{KL}\quad {T_{g}\left\lbrack {\left( {a_{w^{\prime}}{\sum\limits_{w = 1}^{W_{Max}}\quad b_{g,w}^{\prime}}} \right) - \left( {b_{g,w^{\prime}}^{\prime}{\sum\limits_{w = 1}^{W_{Max}}\quad a_{w}}} \right)} \right\rbrack}} = 0} & \left. \left( {\forall{w^{\prime}{{1 \leq w^{\prime} \leq W_{Max}}}}} \right. \right)\end{matrix}{T_{Min} \leq T_{g} \leq {T_{Max}.}}}\quad} & \lbrack 17\rbrack\end{matrix}$

where T_(g) is mask transmission. The constraints in equation [17] forcethe mask Fourier orders to be in the same ratio as the components of theoptimized diffraction orders {right arrow over (a)} from step 2, usingthe order sum as a normalizing factor. A different linear combination ofthe orders with positive coefficients should be used if the componentsof {right arrow over (a)} happen to sum to 0.

The present invention for obtaining an optimized source {right arrowover (s)} and reticle {right arrow over (T)} is not limited to theembodiments described above. Many mathematical techniques other thanthose described above can be brought to bear on this optimizationproblem, as will be clear to those skilled in the art.

The method in accordance with the present invention, as described above,has been embodied in a computer program. The flow diagram for thisembodiment is substantially similar to that outlined in FIG. 5 and FIG.5A, in which step 0.a is performed using a spherical triangulation ofthe bounding hypersphere as described above. FIG. 9 shows the optimizedsource illumination pattern 901 obtained for the desired pattern of FIG.6 obtained using the present embodiment of the computer program inaccordance with the present invention. FIG. 10 shows the optimizedreticle obtained in the present embodiment to print the FIG. 6 patternat a k-factor of 0.38 using chromeless mask technology as is known inthe art in which silicon oxide is etched to different thickness, so thatthere are areas where the mask phase is 0° (1010), and other areas wherethe mask phase is 180° (1020). This example is for the case of F=140 nm,λ=248 nm, NA=0.68. Note that the optimized reticle pattern of FIG. 10bears little resemblance to the FIG. 6 target shapes.

The curved reticle patterns of FIG. 10 are difficult to manufactureusing standard technology. However, it is possible to use a collectionof superimposed rectangular mask shapes (“Manhattan” geometries) toapproximate the optimized diffraction pattern, although with slightlylower efficiency than with the FIG. 10 optimized shapes. FIG. 11 shows areduction of the optimized reticle of FIG. 10 to Manhattan geometries,where the unhatched areas 1110 represent 0° mask phase and the hatchedareas 1120 represent 180° mask phase. The resulting image projected tothe wafer plane is shown in FIG. 12, which can be compared to thedesired feature pattern in FIG. 6 where the edge points 604, 606, 608,610, and 612 are indicated in both figures. The maximum intensitycontour line 1299 on FIG. 12 is approximately at a normalized intensityof 0.45, while the minimum contour line plotted 1211 is approximately ata normalized intensity of 0.025. FIG. 8 shows a plot of the intensityalong lines A-A′, B-B′, C-C′, and D-D′. Contrast is quite high, as isthe linewidth uniformity (comparing intensity at edge points 604, 606,and 608. The intensity contour through the edge points also extends fromtip 610 to tip 612, indicating that line shortening in the aerial imagehas been eliminated. This contour represents the intensity Q in equation[6].

An alternative mask using the optimized source illumination pattern ofFIG. 9 and corresponding optimized diffraction pattern for the desiredFIG. 6 pattern can also be realized. FIG. 13 shows such a reticle,optimized for chrome transmission of 6%. The aerial image issubstantially the same as the image in FIG. 12, except for a reductionin overall intensity, so that the maximum plotted contour 1299 isapproximately at normalized intensity 0.067 while the minimum plottedcontour 1211 is approximately at normalized intensity 0.0033.

FIG. 14 shows an example of a desired feature pattern for anotherembodiment of the present invention, which might represent, for example,the capacitor layout for a dynamic random access memory (DRAM) array.One critical dimension in this pattern is the width 1401 of the brightrectangles 1400, which in this example is equal to 110 nm. Thehorizontal spacing 1402 between features is also 110 nm, thus thehorizontal period is 220 nm. The vertical period (the sum of 1405 and1408 is 330 nm. Though difficult, it is desirable to print therectangles with an aspect ratio (the ratio of length to width 1401 of atleast 1.9:1. FIG. 15 shows the optimized illumination source pattern1501 obtained using the method described above in accordance with thepresent invention for the pattern desired in FIG. 14 using λ=193 nm,NA=0.6, and CD=110 nm.

FIG. 16 shows an optimized reticle using chromeless mask technologywhere unhatched areas 1610 correspond to 0° mask phase areas and hatchedareas 1620 correspond to 180° mask phase areas. FIG. 17 shows areduction of the FIG. 16 reticle to Manhattan geometries where theunhatched areas 1710 correspond to 0° mask phase areas and hatched areas1720 correspond to 180° mask phase areas. The resultant image is shownin FIG. 18, in which the maximum plotted contour 1899 is approximatelyat normalized intensity 0.60 and the minimum plotted contour 1811 isapproximately at normalized intensity 0.05. The bright image featuresare centered on maximum contour 1899, as illustrated by shapes 1499 fromFIG. 14 that are also superimposed on FIG. 18. Again, unlike aconventional mask, the reticle optimized for off-axis illumination bearslittle resemblance to the desired wafer pattern. Note that the brightimage features at 1499 actually fall in between the vaguely brick-likeshapes 1799 from the FIG. 17 mask. In other words, the directresemblance of the reticle shapes 1799 to the target image features 1499is coincidental. The contours 1890 have the desired aspect ratio of1.9:1 corresponding to the aspect ratio of desired rectangular features1499.

FIG. 19 plots the process window attainable with the FIG. 15 and FIG. 17optimized source and reticle. The optimized curve 1903 resulting fromthe source and reticle of the present embodiment is obtained bycalculating aerial images at multiple focal positions, and thencalculating the common exposure range within which both feature lengthand feature width are held within tolerance throughout each plottedfocal range (the horizontal axis coordinate). Tolerances of ±30 nm and±15 nm are used for the length and width, respectively. For comparisonto prior art, a similar curve 1901 is plotted for an image obtained byprior art image enhancement techniques, in which the reticle uses brickshapes corresponding to the desired image features that are fabricatedin attenuating chrome and imaged using annular source illumination.

The integrated area under process window curves like FIG. 19 serves as auseful figure of merit as known in the art for lithographic images (see,for example, R. A. Ferguson, R. M. Marino, and T. A. Brunner, “Dataanalysis methods for evaluating lithographic performance,” J. Vac. Sci.Technol. B 15, no.6 (1997), p. 2387). By this metric the process windowfor the optimized source and reticle of the present invention is 4 timeslarger than with the conventional prior art image enhancement processes.Moreover, at the nominal focus and exposure the optimized processachieves the desired aspect ratio of 1.9:1 with no foreshortening, whilewith the conventional process, the aspect ratio at nominal exposure andfocus is only met within ±30 nm length and ±15 nm width tolerances.

It is difficult to avoid line shortening with prior art enhancementtechniques; thus the process window with the conventional prior arttechnique can be improved if one is willing to compromise the aspectratio. The process window obtained with an attenuating phase shift maskand annular source that are adjusted to print at 1.8:1 aspect ratio isshown by the curve 1902. Though tolerances ease under the relaxed groundrule, the process window for the optimized source and reticle remainslarger than with the conventional enhancement technique by a factor ofabout 2.5, and thus, no compromises need be made in the aspect ratiowhen using image enhancement methods in accordance with the presentinvention.

The optimized reticle patterns obtained in accordance with the presentinvention, whether using chromeless material (for example, FIG. 10 orFIG. 11 corresponding to the pattern in FIG. 6, and FIG. 16 or FIG. 17corresponding to the FIG. 14 pattern) or phase-shifting chrome (forexample, FIG. 13 corresponding to the pattern in FIG. 6), bear littleresemblance to the corresponding desired feature patterns. Recall thatif the desired feature patterns such as FIG. 6 or FIG. 14 are useddirectly to form a mask pattern as in the prior art, the resulting maskpatterns would need to be lengthened to improve aspect ratio (i.e.prevent line shortening), yet would need to be shortened to improvecontrast. Such conflicted shapes are missing from the optimized reticles(FIGS. 10, 11, 13, 16, or 17 obtained using the method in accordancewith the present invention. Prior art reticles (which attempt to conformto the desired image pattern) suffer from problems including leakage oflight from bright features into dark image areas which degradescontrast. By contrast, reticles in accordance with the presentinvention, avoid these problems, and also avoid the need for subsequenttrimming exposures.

Another novel aspect of the particular reticles shown in FIG. 17 is thatthe dark boundaries between adjacent bright features of the image areprinted as the result of destructive interference from phase-shiftedmask openings. In other words, phase cancellation suppresses the imagelight midway between the bright features, forming dark features. Itshould be noted that such phase cancellation action is in no wayprescribed or even identified in the method shown in FIG. 5. accordingto the present invention, and such action does not take place in allsolutions produced by the present invention. However, such action isfound to occur in some optimal reticles such as obtained using theoptimizing process of the present invention as outlined in FIG. 5 (suchas the FIG. 17 reticle). Thus, high quality lithographic images cansometimes be obtained by shaping phase shifted mask openings to printthe dark separations between bright patterns via destructiveinterference. If the mask uses phase-shift chrome, these cancelingshapes can be openings in the chrome. If the mask uses opaque chrome,the phase canceling openings can be etched into the substrate beneaththe chrome, to a depth that puts them 180° out of phase with otheropenings that produce the bright features. Alternatively, phase-shiftedopenings in a chromeless mask can be used. The phase canceling openingsneed not be used to print all dark features in the image; some darkfeatures can instead be printed using such known methods as opaquechrome or phase edges.

The FIG. 5 method in accordance with the present invention will notnecessarily provide masks of this kind, and it is not necessary to usethe FIG. 5 process to design such masks. Instead, specific dimensionsfor the phase canceling features of such masks can be obtained usingdesign processes already known in the art. For example, one can usestandard image simulation methods to calculate the width of dark imagefeatures that are produced by a set of trial phase canceling maskfeatures. The dimensions of these mask features can then be adjusted tocorrect any departures in the widths of the image features from thedesired values. A rule of thumb for choosing initial dimensions for thestarting set of such mask features is that their area be approximately70% that of the corresponding dark regions in the image.

There is a technique (U.S. Pat. No. 5,328,785 entitled “High Power PhaseMasks for Imaging Systems” issued to Smith et al., hereinafter referredto as the Smith patent) which prints dark image areas using a phasegrating composed of alternating bars that are 180° out of phase with oneanother. However, this does not involve the use of phase cancellation toactively suppress the image light between features. Instead, the gratingof the Smith patent works by diffracting all light away from the lenspupil. To achieve this, the phase shifted openings must occupy about 50%of the dark area, and the pitch of the openings must be very fine, forexample less than 0.67λ/NA for the case of illumination with 50% radialpupil fill. As far as image formation is concerned, the lightdistribution produced by the gratings of Smith et al. is the same aswould be produced by ordinary opaque chrome; both block light fromentering the lens. Whether conventional chrome or the gratings of theSmith patent are used, the limited resolution of available lithographylenses causes light from adjacent bright features to spill over into theintervening dark area, distorting the printed shape and degradingcontrast. (The grating has the advantage of being absorption-free, whichprevents mask damage in certain applications.)

Masks are known in the art that use phase shifted chrome in dark areasof the pattern (for example, T. Terasawa et al., “Imagingcharacteristics of multi-phase-shifting and halftone phase-shiftingmasks,” Japanese J. Appl. Phys. Part 1, Vol. 30, no.11B (1991), pages2991-2997, cited above). Phase shift in prior art masks is used tosteepen the transitions at the edges of bright features. However, thetransmission of conventional phase-shifted chrome is fairly low, usuallyless than about 10% that of open areas, and the chrome prints as dark inthe image not because of destructive interference, but simply because itblocks light in the same way as does ordinary chrome (though not quiteas completely). Another difference between the above-described masks ofthe present invention and the prior art masks of Terasawa et al. is thatchrome is not physically opened in these prior art masks to print darkregions; instead, mask openings are used to print bright image regions.Yet another difference is that the chrome used in these prior art masksdoes not constitute a designed set of shapes in the mask; instead, it issimply a continuous surround (i.e. background) for the designed openshapes that print as bright image regions.

Masks such as shown in FIG. 17 of this aspect of the present invention,employ phase shifted openings of designed shape to form dark separationsbetween bright image features by means of destructive interference toachieve superior resolution in comparison to a prior art chrome mask orto a grating mask of the Smith patent. Resolution is improved becausethe phase-shifted light is deliberately diffracted into the lens pupilby the phase canceling features, rather than being diffracted away as inthe Smith patent. This phase-shifted light cancels some of theabove-mentioned spillover light, improving resolution and contrast. Thephase-shifted mask openings are laid out periodically only if thedesired image features are periodic. In the FIG. 17 example the y imagepitch (660 nm) is about 2.05λ/NA; this is large enough that light fromthe phase-shifted openings is collected by the lithographic lens,allowing it to produce dark features in the image by destructiveinterference.

While the invention has been described in terms of specific embodiments,it is evident in view of the foregoing description that numerousalternatives, modifications and variations will be apparent to thoseskilled in the art. Accordingly, the invention is intended to encompassall such alternatives, modifications and variations which fall withinthe scope and spirit of the invention and the following claims.

We claim:
 1. A method for printing a desired pattern on a semiconductorwafer having a photoactive material thereon, the method comprising thesteps of: providing a diffraction relationship for determining at leastone diffracted amplitude diffracted by a reticle having at least onediffraction parameter when said reticle is exposed to illuminationenergy from an illumination source having at least one source parameter;providing a merit function for computing a merit value of an imagepattern on a wafer plane as a function of variables including said atleast one diffracted amplitude; selecting at least one constraint on atleast one of said variables; determining an optimal value of said atleast one diffracted amplitude so that said merit function attains amerit value that is optimized in accordance with said at least oneconstraint; selecting a combination of at least one selected sourceparameter and at least one selected diffraction parameter to producesaid optimal value of said at least one diffracted amplitude inaccordance with said diffraction relationship; forming an optimalreticle having a diffractive feature with said at least one selecteddiffraction parameter; illuminating said optimal reticle withillumination energy from an illumination source having said at least oneselected source parameter, so that said illumination energy isdiffracted by said diffractive feature and projected through a lensaperture to form an optimal image pattern on the wafer plane inaccordance with said at least one constraint; exposing the photoactivematerial to said optimal image pattern; and developing said exposedphotoactive material to form the desired pattern.
 2. The method of claim1 wherein said at least one source parameter comprises both a sourcedirection and a source intensity.
 3. The method of claim 1 furthercomprising forming an optimal reticle having at least one diffractivefeature with said at least one selected diffraction parameter so thatsaid optimal value of said at least one diffracted amplitude is producedwhen said diffractive feature is illuminated by an illumination sourcehaving said at least one selected source parameter.
 4. The method ofclaim 1 wherein said at least one diffracted amplitude further comprisesa first amplitude collected within a first quadrant of the lens apertureand a second amplitude collected within a second quadrant of the lensaperture, wherein said first amplitude is selected independently of saidsecond amplitude.
 5. The method of claim 1 wherein said at least oneconstraint includes a preselected image intensity at a selected point ofthe desired image pattern corresponding to an edge of an element of thedesired image pattern.
 6. The method of claim 1 wherein said meritfunction comprises a derivative of image intensity at a selected pointon the wafer plane corresponding to an edge of an element of the desiredimage pattern, said derivative having a direction normal to said edge.7. A method for printing an integrated circuit pattern on asemiconductor wafer having a photoactive material thereon, the methodcomprising the steps of: providing a desired wafer feature patternhaving at least one wafer feature element; deriving a merit functiondescribing a relationship between an illumination source, a reticle, andan image, said source having at least one source parameter, said reticlehaving at least one diffractive feature, and said image having at leastone image intensity; selecting at least one constraint in relation tosaid desired wafer feature pattern that said at least one imageintensity must satisfy; selecting a combination of said at least onesource parameter and said at least one diffractive feature so that saidmerit function is optimized in accordance with said at least oneconstraint; illuminating said reticle with illumination energy from saidillumination source, so that said illumination energy is diffracted bysaid reticle and projected through a lens aperture to form said at leastone image intensity on the wafer; exposing the photoactive material tosaid at least one image intensity; and developing said exposedphotoactive material to form a printed feature, so that said printedfeature conforms with said at least one wafer feature element of saiddesired wafer feature pattern in accordance with said at least oneconstraint, and wherein said at least one source parameter comprises asource direction having a source amplitude, wherein illumination fromsaid source direction on said at least one diffractive feature producesat least one diffracted amplitude, and said step of selecting acombination further comprises selecting a plurality of source directionshaving diffracted orders overlapping in direction space in accordancewith said desired wafer feature pattern; deriving a simplified functionfrom said merit function; computing a diffracted amplitude correspondingto each of said plurality of source directions at selected sourceamplitudes and selecting a first optimized combination of computeddiffracted amplitudes, source directions and source amplitudes so thatsaid simplified function is globally optimized; selecting a finaloptimized combination of at least one source direction, at least onesource amplitude and at least one diffracted amplitude so that saidmerit function is locally optimized using said first optimizedcombination as a starting solution, and so that said merit function islocally optimized in accordance with said at least one constraint; andforming said selected at least one diffractive feature so that said atleast one selected diffracted amplitude is produced when said selectedat least one diffractive feature is illuminated by said source from saidselected at least one source direction.
 8. The method of claim 7 whereinthe step of forming said selected at least one diffractive featurefurther comprises approximating said diffractive feature usingsuperimposed rectangular shapes.
 9. A method for selecting a combinationof source illumination parameters and diffraction mask features forprojecting energy through a lens aperture to form a desired imagepattern on a wafer plane, the method comprising the steps of: providinga diffraction relationship for determining at least one diffractedamplitude diffracted by a reticle having at least one diffractionparameter when said reticle is exposed to illumination energy from anillumination source having at least one source parameter; providing amerit function for computing a merit value of the desired image patternon the wafer plane as a function of variables including said at leastone diffracted amplitude; selecting at least one constraint on at leastone of said variables; determining an optimal value of said at least onediffracted amplitude so that said merit function attains a merit valuethat is optimized in accordance with said at least one constraint; andselecting a combination of at least one selected source parameter and atleast one selected diffraction parameter to produce said optimal valueof said at least one diffracted amplitude in accordance with saiddiffraction relationship.
 10. The method of claim 9 wherein said atleast one source parameter comprises both a source direction and asource intensity.
 11. The method of claim 9 further comprising designinga reticle having a diffractive feature with said at least one selecteddiffraction parameter so that said optimal value of said at least onediffracted amplitude is produced when said diffractive feature isilluminated b~ an illumination source having said at least one selectedsource parameter.
 12. The method of claim 9 wherein said at least onediffracted amplitude further comprises a first amplitude collectedwithin a first quadrant of the lens aperture and a second amplitudecollected within a second quadrant of the lens aperture, wherein saidfirst amplitude is selected independently of said second amplitude. 13.The method of claim 9 wherein said at least one constraint includes apreselected image intensity at a selected point of the desired imagepattern corresponding to an edge of an element of the desired imagepattern.
 14. The method of claim 9 wherein said merit function comprisesa derivative of image intensity at a selected point on the wafer planecorresponding to an edge of an element of the desired image pattern,said derivative having a direction normal to said edge.
 15. A method forselecting a combination of source illumination parameters anddiffraction mask features for projecting energy through a lens apertureto form an image pattern on a wafer, the method comprising the steps of:providing a desired wafer feature pattern having at least one waferfeature element; deriving a merit function describing a relationshipbetween an illumination source, a reticle, and the image pattern, saidsource having at least one source parameter, said reticle having atleast one diffractive feature, and said image pattern having at leastone image intensity; selecting at least one constraint in relation tosaid desired wafer feature pattern that said at least one imageintensity must satisfy; and selecting a combination of said at least onesource parameter and said at least one diffractive feature, so that saidmerit function is optimized in accordance with said at least oneconstraint, and wherein said at least one source parameter comprises asource direction having a source amplitude, wherein illumination fromsaid source direction on said at least one diffractive feature producesat least one diffracted amplitude and said step of selecting acombination further comprises: selecting a plurality of sourcedirections having diffracted orders overlapping in direction space inaccordance with said desired wafer feature pattern; deriving asimplified function from said merit function; computing a computeddiffracted amplitude corresponding to each of said plurality of sourcedirections at selected source amplitudes and selecting a first optimizedcombination of computed diffracted amplitudes, source directions andsource amplitudes so that said simplified function is globallyoptimized; selecting a final optimized combination of at least onesource direction, at least one source amplitude and at least onediffracted amplitude so that said merit function is locally optimizedusing said first optimized combination as a starting solution, and sothat said merit function is locally optimized in accordance with said atleast one constraint; and forming said selected at least one diffractivefeature so that said at least one selected diffracted amplitude isproduced when said selected at least one diffractive feature isilluminated by said source from said selected at least one sourcedirection.
 16. A computer program product for selecting a combination ofsource illumination parameters and diffraction mask features forprojecting energy through a lens aperture to form a desired image, thecomputer program product comprising computer readable instructions forcausing a computer to perform the method steps of: storing a desiredimage pattern on a wafer plane; storing a diffraction relationship fordetermining at least one diffracted amplitude diffracted by a reticlehaving at least one diffraction parameter when said reticle is exposedto illumination energy from an illumination source having at least onesource parameter; storing a merit function for computing a merit valueof an image pattern on the wafer plane as a function of variablesincluding said at least one diffracted amplitude; storing at least oneconstraint on at least one of said variables; determining an optimalvalue of said at least one diffracted amplitude so that said meritfunction attains a merit value that is optimized in accordance with saidat least one constraint; and selecting a combination of at least oneselected source parameter and at least one selected diffractionparameter to produce said optimal value of said at least one diffractedamplitude in accordance with said diffraction relationship.
 17. Thecomputer program product of claim 16 wherein said at least one sourceparameter comprises both a source direction and a source intensity. 18.The computer program product of claim 16 wherein said method stepsfurther comprise computing characteristics of a reticle having adiffractive feature with said at least one selected diffractionparameter so that said optimal value of said at least one diffractedamplitude is produced when said diffractive feature illuminated by anillumination source having said at least one selected source parameter.19. The method of claim 16 said at least one diffracted amplitudefurther comprises a first amplitude collected within a first quadrant ofthe lens aperture and a second amplitude collected within a secondquadrant of the lens aperture, wherein said first amplitude is selectedindependently of said second amplitude.
 20. The computer program productof claim 16 wherein said at least one constraint includes a preselectedimage intensity at a selected point of the desired image patterncorresponding to an edge of an element of the desired image pattern. 21.The computer program product of claim 16 wherein said merit functioncomprises a derivative of image intensity at a selected point on thewafer plane corresponding to an edge of an element of the desired imagepattern, said derivative having a direction normal to said edge.
 22. Acomputer program product for selecting a combination of sourceillumination parameters and diffraction mask features for projectingenergy through a lens aperture to form a desired image, the computerprogram product comprising computer readable instructions for causing acomputer to perform a method having the steps of: storing a desiredwafer pattern having at least one wafer feature element; storing a meritfunction describing a relationship between an illumination source, areticle, and an image pattern, said source having at least one sourceparameter, said reticle having at least one diffractive feature, andsaid image pattern having at least one image intensity; storing at leastone constraint in relation to said desired wafer feature pattern thatsaid at least one image intensity must satisfy; and selecting acombination of said at least one source parameter and said at least onediffractive feature, so that said merit function is optimized inaccordance with said at least one constraint, and wherein said at leastone source parameter comprises a source direction having a sourceamplitude, wherein illumination from said source direction on said atleast one diffractive feature produces at least one diffracted amplitudeand said step of selecting a combination further comprises: storing aplurality of source directions having diffracted orders overlapping indirection space in accordance with said desired feature pattern;computing a simplified function derived from said merit function;computing a diffracted amplitude corresponding to each of said pluralityof source directions at selected source amplitudes and selecting a firstoptimized combination of computed diffracted amplitudes, sourcedirections and source amplitudes so that said simplified function isglobally optimized; selecting a final optimized combination of at leastone source direction, at least one source amplitude and at least onediffracted amplitude so that said merit function is locally optimizedusing said first optimized combination as a starting solution, and sothat said merit function is locally optimized in accordance with said atleast one constraint; and computing characteristics of said selected atleast one diffractive feature so that said at least one selecteddiffracted amplitude is produced when said selected at least onediffractive feature is illuminated by said source from said selected atleast one source direction.
 23. The method of claim 22 wherein the stepof computing characteristics of said selected at least one diffractivefeature further comprises approximating said diffractive feature usingsuperimposed rectangular shapes.
 24. A machine readable storage mediumhaving stored therein a program of instructions executable by themachine to perform method steps for selecting a combination of sourceillumination parameters and diffraction mask features for projectingenergy through a lens aperture to form a desired image pattern on awafer plane, said method steps comprising: storing a desired imagepattern; storing a diffraction relationship for determining at least onediffracted amplitude diffracted by a reticle having at least onediffraction parameter when said reticle is exposed to illuminationenergy from an illumination source having at least one source parameter;storing a merit function for computing a merit value of an image patternon the wafer plane as a function of variables including said at leastone diffracted amplitude; storing at least one constraint on at leastone of said variables; determining an optimal value of said at least onediffracted amplitude so that said merit function attains a merit valuethat is optimized in accordance with said at least one constraint; andselecting a combination of at least one selected source parameter and atleast one selected diffraction parameter to produce said optimal valueof said at least one diffracted amplitude in accordance with saiddiffraction relationship.
 25. The machine readable storage medium ofclaim 24 wherein said at least one source parameter comprises both asource direction and a source intensity.
 26. The machine readablestorage medium of claim 24 wherein said method steps further comprisecomputing characteristics of a reticle having a diffractive feature withsaid at least one selected diffraction parameter so that said optimalvalue of said at least one diffracted amplitude is produced when saiddiffractive feature is illuminated by an illumination source having saidat least one selected source parameter.
 27. The method of claim 24wherein said at least one diffracted amplitude further comprises a firstamplitude collected within a first quadrant of the lens aperture and asecond amplitude collected within a second quadrant of the lensaperture, wherein said first amplitude is selected independently of saidsecond amplitude.
 28. A machine readable storage medium having storedtherein a program of instructions executable by the machine to performmethod steps for selecting a combination of source illuminationparameters and diffraction mask features for projecting energy through alens aperture to form a desired image, said method steps comprising:storing a desired wafer feature pattern having at least one waferfeature element; storing instructions for causing a computer to computea merit function describing a relationship between an illuminationsource, a reticle, and an image pattern, said source having at least onesource parameter, said reticle having at least one diffractive feature,and said image pattern having at least one image intensity; storing atleast one constraint in relation to said desired feature pattern thatsaid at least one image intensity must satisfy; and selecting acombination of said at least one source parameter and said at least onediffractive feature, so that said merit function is optimized inaccordance with said at least one constraint, and wherein the step ofselecting a combination further comprises approximating said diffractivefeature using superimposed rectangular shapes.
 29. The machine readablestorage medium of claim 24 wherein said at least one constraint includesa preselected image intensity at a selected point of the desired imagepattern corresponding to an edge of an element of the desired imagepattern.
 30. The machine readable storage medium of claim 24 whereinsaid merit function comprises a derivative of image intensity at aselected point of the desired image pattern corresponding to an edge ofan element of the desired image pattern, said derivative having adirection normal to said edge.
 31. A machine readable storage mediumhaving stored therein a program of instructions executable by themachine to perform method steps for selecting a combination of sourceillumination parameters and diffraction mask features for projectingenergy through a lens aperture to form a desired image, said methodsteps comprising: storing a desired wafer feature pattern having atleast one wafer feature element; storing instructions for causing acomputer to compute a merit function describing a relationship betweenan illumination source, a reticle, and an image pattern, said sourcehaving at least one source parameter, said reticle having at least onediffractive feature, and said image pattern having at least one imageintensity; storing at least one constraint in relation to said desiredfeature pattern that said at least one image intensity must satisfy; andselecting a combination of said at least one source parameter and saidat least one diffractive feature, so that said merit function isoptimized in accordance with said at least one constraint, and whereinsaid at least one source parameter comprises a source direction having asource amplitude, wherein illumination from said source direction onsaid at least one diffractive feature produces at least one diffractedamplitude and said step of selecting a combination further comprises:selecting a plurality of source directions having diffracted ordersoverlapping in direction space in accordance with said desired waferfeature pattern; deriving a simplified function from said meritfunction; computing a diffracted amplitude corresponding to each of saidplurality of source directions at selected source amplitudes andselecting a first optimized combination of computed diffractedamplitudes, source directions and source amplitudes so that saidsimplified function is globally optimized; selecting a final optimizedcombination of at least one source direction, at least one sourceamplitude and at least one diffracted amplitude so that said meritfunction is locally optimized using said first optimized combination asa starting solution, and so that said merit function is locallyoptimized in accordance with said at least one constraint; and designingsaid selected at least one diffractive feature so that said selected atleast one diffracted amplitude is produced when said selected at leastone diffractive feature is illuminated by said source from said selectedat least one source direction.
 32. A machine readable storage mediumhaving stored therein a program of instructions executable by themachine to perform method steps for selecting a combination of sourceillumination parameters and diffraction mask features for projectingenergy through a lens aperture to form a desired image, said methodsteps comprising: storing a desired wafer feature pattern having atleast one wafer feature element; storing instructions for causing acomputer to compute a merit function describing a relationship betweenan illumination source, a reticle, and an image pattern, said sourcehaving at least one source parameter, said reticle having at least onediffractive feature, and said image pattern having at least one imageintensity; storing at least one constraint in relation to said desiredfeature pattern that said at least one image intensity must satisfy; andselecting a combination of said at least one source parameter and saidat least one diffractive feature, so that said merit function isoptimized in accordance with said at least one constraint, and whereinthe step of selecting a combination further comprises approximating saidselected at least one diffractive feature using superimposed rectangularshapes.
 33. A lithographic system for printing a desired wafer featurepattern on a semiconductor wafer including a photoactive material, thesystem comprising: an illumination source having at least one sourceparameter; a reticle having at least one diffraction parameter; and alens; said illumination source, said reticle and said lens beingarranged so that said illumination source illuminates said reticle so asto produce a plurality of diffracted amplitudes and said plurality ofdiffracted amplitudes are collected by said lens and projected to form aimage on the semiconductor wafer and wherein said at least one sourceparameter and said at least one diffraction parameter are selected sothat a merit function, expressed as a function of variables includingsaid plurality of diffracted amplitudes, attains an optimal merit valueand wherein said merit function is optimized in accordance with at leastone constraint on at least one of said variables, so that exposing thephotoactive material to said image and developing said exposedphotoactive material forms a printed pattern that substantially conformswith the desired wafer feature pattern.
 34. A reticle for diffractingillumination energy to form a desired image pattern having a pattern ofintensities, the desired image pattern having a desired bright area inwhich the intensities within the desired bright area exceed apredetermined bright threshold and having a desired dark area in whichthe intensities within the desired dark area are less than apredetermined dark threshold, the reticle comprising a pattern ofphase-shifting material arranged so that the desired dark area is formedcompletely by destructive interference of diffracted illumination energydiffracted by said pattern of phase-shifting material.
 35. The reticleof claim 34 wherein said phase-shifting material comprisesphase-shifting chrome material that transmits at least some illuminationenergy.
 36. The method of claim 1 wherein said at least one sourceparameter comprises a source direction having a source amplitude, andsaid step of selecting a combination further comprises: selecting aplurality of source directions having diffracted orders overlapping indirection space in accordance with the desired image pattern; deriving asimplified function from said merit function; computing a diffractedamplitude corresponding to each of said plurality of source directionshaving source amplitudes and selecting a first optimized combination ofcomputed diffracted amplitudes, source directions and source amplitudesso that said simplified function is globally optimized; selecting afinal optimized combination of at least one selected source directionhaving a selected source amplitude and at least one selected diffractedamplitude so that said merit function is locally optimized using saidfirst optimized combination as a starting solution, and so that saidmerit function is locally optimized in accordance with said at least oneconstraint; and forming said diffractive feature so that said at leastone selected diffracted amplitude is produced when said diffractivefeature is illuminated by an illumination source having said at leastone selected source direction having said selected source amplitude. 37.The method of claim 1 wherein the step of forming said optimal reticlefurther comprises approximating said diffractive feature usingsuperimposed rectangular shapes.
 38. The method of claim 9 wherein saidat least one source parameter comprises a source direction having asource amplitude and said step of selecting a combination furthercomprises: selecting a plurality of source directions having diffractedorders overlapping in direction space in accordance with the desiredimage pattern; deriving a simplified function from said merit function;computing a computed diffracted amplitude corresponding to each of saidplurality of source directions at selected source amplitudes andselecting a first optimized combination of computed diffractedamplitudes, source directions and source amplitudes so that saidsimplified function is globally optimized; selecting a final optimizedcombination of at least one selected source direction having a selectedsource amplitude and at least one selected diffracted amplitude so thatsaid merit function is locally optimized using said first optimizedcombination as a starting solution, and so that said merit function islocally optimized in accordance with said at least one constraint; andforming a reticle having a diffractive feature with said at least oneselected diffraction parameter so that said at least one selecteddiffracted amplitude is produced when said diffractive feature isilluminated by an illumination source having said at least one selectedsource direction having said selected source amplitude.
 39. The computerprogram product of claim 16 wherein said at least one source parametercomprises a source direction having a source amplitude, and said step ofselecting a combination further comprises: storing a plurality of sourcedirections having diffracted orders overlapping in direction space inaccordance with said desired feature pattern; computing a simplifiedfunction derived from said merit function; computing a diffractedamplitude corresponding to each of said plurality of source directionsat selected source amplitudes and selecting a first optimizedcombination of computed diffracted amplitudes, source directions andsource amplitudes so that said simplified function is globallyoptimized; selecting a final optimized combination of at least oneselected source direction having a selected source amplitude and atleast one selected diffracted amplitude so that said merit function islocally optimized using said first optimized combination as a startingsolution, and so that said merit function is locally optimized inaccordance with said at least one constraint; and computingcharacteristics of a reticle having a diffractive feature with said atleast one selected diffraction parameter so that said at least oneselected diffracted amplitude is produced when said diffractive featureis illuminated by an illumination source having said at least oneselected source direction having said selected source amplitude.
 40. Themethod of claim 16 wherein the computer readable instructions forcomputing characteristics of said optimal reticle further comprisesapproximating said diffractive feature using superimposed rectangularshapes.
 41. The machine readable storage medium of claim 24 wherein saidat least one source parameter comprises a source direction having asource amplitude, and said step of selecting a combination furthercomprises: selecting a plurality of source directions having diffractedorders overlapping in direction space in accordance with said desiredwafer feature pattern; deriving a simplified function from said meritfunction; computing a diffracted amplitude corresponding to each of saidplurality of source directions at selected source amplitudes andselecting a first optimized combination of computed diffractedamplitudes, source directions and source amplitudes so that saidsimplified function is globally optimized; selecting a final optimizedcombination of at least one selected source direction having a selectedsource amplitude and at least one selected diffracted amplitude so thatsaid merit function is locally optimized using said first optimizedcombination as a starting solution, and so that said merit function islocally optimized in accordance with said at least one constraint; andcomputing characteristics of said optimal reticle having saiddiffractive feature so that said at least one selected diffractedamplitude is produced when said diffractive feature is illuminated by anillumination source having said at least one selected source directionhaving said selected source amplitude.
 42. The machine readable storagemedium of claim 26 wherein the step of computing characteristics of areticle further comprises approximating said diffractive feature usingsuperimposed rectangular shapes.
 43. The method of claim 11 wherein thestep of designing a reticle further comprises approximating saiddiffractive feature using superimposed rectangular shapes.